# Integral equations, spring 2015

### Lecturers

Miren Zubeldia and Rodrigo Bleyer

### Scope

5 + 5 sp.

### Type

Advanced studies

### Prerequisites

Familiarity with classical real analysis, basic linear algebra and rudiments of ordinary differential equation theory.

### Description

This course is divided into two parts:

__Part one:__ This part will be an introduction to linear integral equations, that is, equations involving an unknown function which appears under an integral sign (and where the dependence on this function is linear). Such equations occur widely in diverse areas of applied mathematics and physiscs and they offer a powerful technique for solving a variety of practical problems. In particular, we will focus on their connection to partial differential equations and we will emphasize their fundamental role in initiating the development of functional analysis as the appropriate abstract framework for studying integral and also differential equations.

Table of contents will be as follows:

- Volterra equations
- Bounded and compact operators in a Hilbert space
- Riesz theory for compact operators
- Fredholm integral equations
- Fredholm alternative and Fredholm operators
- Spectral theory for linear compact self-adjoint operators

__Part two:__ This part we will be an quick survey to numerical solution to linear integral equations of the second part, seen in the first part. The main topics will be studied individually and presented to the entire group as a seminar.

Table of contents will be as follows:

- Quadrature methods (Nyström)
- Projection methods (Collocation and Galerkin)
- Iteration methods (two-grid and multi-grid)
- Boundary integral equation and finite element methods (FEM)

### Lectures

__Part one:__ Weeks 3-9, Tuesday 10-12 and Thursday 10-12 in room B322. In addition, there will be 2 hours of exercises every week.

__Part two:__ Tuesday 10-12 and Thursday 10-12 in room B322; Wednesday 14-16 in room C122

- Weeks 10 - 11 Reading material
- Week 12 Individual meeting
- Weeks 13 - 16 Development of seminar and numerical implementation
- Weeks 17 - 18 Seminar presentation

Easter Holiday 2.-8.4.

Lecture Notes: integral_equations_notes.pdf

### Exam

__Part one:__ Feb 25, 2015 (Wednesday 14-16 room C122)

__Part two:__ Seminar presentation and report.

### Bibliography

__Part one:__

- R. Kress. Linear Integral Equations, Applied Mathematical Sciences, Volume 82, Springer-Verlag, 1989.
- K. Yosida. Lectures on Differential and Integral Equations, Dover Publications, INC., New York, 1991.
- D. Porter, D.S.G. Stirling. Integral equations a practical treatment, from spectral theory to applications, Cambridge Texts in Applied Mathematics, 1990.

__Part two:__

- Kendall E. Atkinson. The numerical solution of integral equations of the second kind, Cambridge University Press, 2009.

### Registration

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### Exercises

Group | Day | Time | Place | Instructor |
---|---|---|---|---|

1. | Wednesday | 14-16 | C122 | Rodrigo Bleyer |

### Exercise List

- List 1 (due to 21.01.2015) >> ExList1.pdf <<
- List 2 (due to 28.01.2015) >> ExList2.pdf <<
- List 3 (due to 04.02.2015) >> ExList3.pdf <<
- List 4 (due to 11.02.2015) >> ExList4.pdf <<
- List 5 (due to 18.02.2015) >> ExList5.pdf <<